Finite Group Databasebeta

← #30640694

Subgroups — #30640694

Order 16256.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550311C211yesmaximal, characteristic, complemented
2C2×#16861152C211yesmaximal, characteristic, complemented
3#22550146C211yesmaximal, characteristic, complemented
4#22549692C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#578×C127C211yesmaximal, characteristic, complemented
8#16861234—21no—
9C2×D1016C2211yescharacteristic
10#16861152—22no—
11#16861152—21no—
12#16861150—21no—
13#16861149C2211yescharacteristic
14C2×C127⋊C16C2211yescharacteristic
15#16861145C2211yescharacteristic
16#16861125—21no—
172−1+4×C127—21no—
18#137×C127—22no—
19#137×C127—21no—
20#135×C127—21no—
21#135×C127C2211yescharacteristic
22C2×Q16×C127—21no—
23C2×Q16×C127C2211yescharacteristic
24#131×C127C2211yescharacteristic
25C2×D508—21no—
26#14523235—41no—
27D1016—21no—
28D1016—21no—
29#14523233—41no—
30#14523232—21no—
31C127⋊C16—21no—
32C127⋊C16—21no—
33#41×C127—22no—
34#41×C127—21no—
35#41×C127—42no—
36#41×C127—41no—
37#41×C127D411yescharacteristic
38C2×Q8×C127—22no—
39C2×Q8×C127—21no—
40C2×Q8×C127—41no—
41Q16×C127—22no—
42Q16×C127—21no—
43Q16×C127D411yescharacteristic
44Q16×C127D412yes—
45SD16×C127—22no—
46SD16×C127—21no—
47D8×C127D411yescharacteristic
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52C22×D127—41no—
53D508—41no—
54D508—41no—
55C127⋊C8—41no—
56Q8×C127—22no—
57Q8×C127—21no—
58Q8×C127—42no—
59Q8×C127—41no—
60Q8×C127D811yescharacteristic
61D4×C127—21no—
62D4×C127—21no—
63D4×C127—42no—
64D4×C127—41no—
65D4×C127D811yescharacteristic
66C2×C4×C127—21no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—21no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—41no—
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—81no—
79C2×D127—161no—
80C4×C127—21no—
81C4×C127—42no—
82C4×C127—41no—
83C4×C127#12011yescharacteristic
84C4×C127C2×D811yescharacteristic
85C2×C127—21no—
86C2×C127—41no—
87C2×C127—81no—
88C2×C127#44711yescharacteristic
89D127—161no—
90#2193—1271nomaximal, Sylow
91C127#219311yescharacteristic, complemented, Sylow
92#578D12711yescharacteristic, complemented
93#509—1271no—
94C2×SD32—1271no—
95#472—1271no—
96#362—1271no—
97#359—1271no—
98#350—1271no—
992−1+4—21no—
100#137—22no—
101#137—21no—
102AGL(1,ℤ/8)—2541no—
103#135—21no—
104#135C2×D12711yescharacteristic
105C2×Q16—21no—
106C2×Q16C2×D12711yescharacteristic
107C2×D8—1271no—
108#131C2×D12711yescharacteristic
109SD32—2542no—
110SD32—2541no—
111D16—2541no—
112M32—1271no—
113C2×C16—1271no—
114#108—1271no—
115#100—2541no—
116#41—22no—
117#41—21no—
118#41—42no—
119#41—41no—
120#41C22⋊D12711yescharacteristic
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—41no—
124C2×D4—2541no—
125Q16—22no—
126Q16—21no—
127Q16C22⋊D12711yescharacteristic
128Q16C22⋊D12712yes—
129SD16—22no—
130SD16—21no—
131SD16—5081no—
132D8—5081no—
133D8—2541no—
134D8—2541no—
135D8C22⋊D12711yescharacteristic
136M16—21no—
137M16—2541no—
138M16C22⋊D12711yescharacteristic
139C2×C8—21no—
140C2×C8C22×D12711yescharacteristic
141C16—2541no—
142C16—2541no—
143C23—5081no—
144Q8—22no—
145Q8—21no—
146Q8—42no—
147Q8—41no—
148Q8#1452323311yescharacteristic
149D4—21no—
150D4—21no—
151D4—42no—
152D4—41no—
153D4—5081no—
154D4—5081no—
155D4#1452323311yescharacteristic
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×C22⋊D12711yescharacteristic
162C8—21no—
163C8—5081no—
164C8C2×C22⋊D12711yescharacteristic
165C8C2×C22⋊D12711yescharacteristic
166C22—21no—
167C22—41no—
168C22—20321no—
169C22—10161no—
170C22#1686123511yescharacteristic
171C4—21no—
172C4—42no—
173C4—41no—
174C4#1686117811yescharacteristic
175C4C2×#1452323311yescharacteristic
176C2—21no—
177C2—41no—
178C2—81no—
179C2—20321no—
180C2#2255001011yescharacteristic
subgroupG/H
centerC2#22550010
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#578×C127C2
socleC2×C127#447
hypercenter#578D127
hypocenterC127#2193
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #578

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550311C2semidirect
2. C2×#16861152C2semidirect
3. #22550146C2semidirect
4. #22549692C2semidirect
5. #22549683C2semidirect
6. #22549660C2semidirect
7. #578×C127C2semidirect
91. C127#2193semidirect
92. #578D127semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225503112
C2×#168611522
#225501462
#225496922
#225496832
#225496602
#578×C1272
#2193127